International Journal of Scientific Research and Management (IJSRM)
||Volume||08||Issue||05||Pages||EC-2020-386-396||2020||
Website: www.ijsrm.in ISSN (e): 2321-3418 DOI: 10.18535/ijsrm/v8i05.ec02
Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and
Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”
1Dr. Amresh Kumar*, 2Dr. Ram Kishore Singh
1Assistant professor, Dept-ECE, RRSDCE, Begusarai,
2Associate Professor (Retd.), Dept- Electrical Engg., MIT, Muzaffarpur Abstract
Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of
"Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class.
Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable.
Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.
Keywords: Hilbert Spaces, Differential Equations, Fourier Transform Orthogonal Polynomials, Optimization and Approximation Theory, Signal Processing On 2- Sphere
1.1 Introduction
Since its introduction in communication engineering by Shannon, the sampling theorem has played an important role in both mathematics and electrical engineering. The sampling theorem has derived for functions with valued in separable Hilbert space. The proof uses the concept of frames and frame operators
in Hilbert space. One of the advantages of this theorem is that it allows us to derive sampling theorem associated with bounding values problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem.
The theorem says that if a signal (function) of time, t, is limited in its band width to W cycles/ second, it is completely determined by its values at a series of discrete point spaced 1/2W seconds apart and can be recovered at any time t by using its values at discrete set of points.
Mathematically this can be represented as If f(t) is a function band limited to [-
- (1.1)
-
k = 0, ±1, ±2, ... via the formula
- - - (1.2)
Where the series converges absolutely and uniformly on compact set of the real line. This theorem has been generalized in many different ways.
[1] -
uniformaly spaced points leading to the following generalization by Paley and W sequence of real numbers such that
– (1.3)
And let P(t) be the entire function defined by
P (t) = (t – – (t/tk)) (1 – (t/t-k)) (1.4) Then for any f in the form equation (6.1) we have
f ( - -tk) P'tk } (1.5)
[2] In another direction, the
I = [a,b], -
orthogonal family of function in L2(I). Then for any f of the form
(1.6) Where
- (1.7)
Where
—(—x—,—t— (1.8)
If I = [-
- -tk)
and hence equation (1.7) reduces to equation (1.2).
[3] -
Lioville bounding value problem.
-y" + q(x) y = (1.9)
(1.10)
(1.11)
where q is continuous on I. Then take K(x,t) to be the solution of the differential equation (1.9) and the initial condition (1.10) or the solution of (1.9) and (1.10) and take t
Although it is theoretically feasible to extend this procedure to more general self-adjoint boundary value problem associated with the nth order differential operators in practice this does not always work since existence of one single function K(x,t) which generates all the eigen functions of the problem when parameter t is replaced by the eigen values, is not always guaranted. For example
-
are not generated by one single real-valued function. One possibility to circumvent this problem is to use the Green's function method described in previous chapter.
For many self-adjoint boundary value problems, the green functions can be written as
- (1.12)
function method can also be used to derive sampling theorem associated with homogeneous.
The aim of the present thesis is to generalize some of the above results obtained to derive a sampling theorem for vector-valued functions. These functions take values in a separable Hilbert space H. One of the interesting ramifications of this generalisation is that allows us to obtain sampling theorem associated with boundary value problems and integral equations without restricting the sampling point to be the eigen values of the corresponding problem. In fact, the sampling poins will be arbitrary, except for a restriction on their growth rate. The idea of using Hilbert space concept in sampling theory used the concept of frames in a Hilbert space to derive sampling theorems for band-limited functions. In this chapter we also derive our sampling theorem using the concept of frames.
1.2 Preliminaries
Throughout the thesis the sets of complex and real numbers have been denoted by C and respectively and we have denote
- (1.13)
so that the inverse transform is given by
- - (1.14)
Provided that the integral exist.
real axis
(1.15) where Z = x + iy and
- (1.16)
The well-known Paley-
- (1.17)
- -Wiener class of entire functions.
Let G = {gn} be a sequence in Hilbert space . We say that G is a frame if there exist two number A, B > 0
(1.18)
The two numbers A and B are called frame bounds. The frame is said to be light if A = B and is exact if it ceases to be a frame whenever any single element is detected from the frame. Frames are complete if <f, gn>
= 0, for all n, then ||f|| = 0 and consequently f = 0, we have already derived in previous section, G is said to
be bounded if there exist two non- l n. It is known
that a frame is exact if and only if it is a bounded and unconditional basis. A basis G is said to be unconditional if
(1.19) with every G, we associate a frame operator S defined by
Sf = (1.20)
The inverse frame operator S-1 has the following properties
[1] B- - -1I, and
[2] {S-1gn} is also frame with frame bounds B-1 and A-1.
1.3 The main result:
(1.20)
) converges.
Also let,
–
(1.21) –
–
(1.22) –
separable Hilbert space H and S be its frame operator. Then the dual frame {S-
0 i
(1.23) 1 if m = n
- (1.24)
Then the above operator applied on Hilbert space H in the usual way we have
- (1.25)
(1.26)
therefore (1.27) Having basic ideas we are in a position to prove sampling theorem.
mpact subset of the -
[2]
and can be reconstructed from these values according to the formula.
-
h > | (1.28)
Let - (1.29)
-Schwarz inequality as explained in previous chapter. We have
– – (1.30)
-
-
(1.31)
-1 and A-1, yield
– - -1
(1.32)
- —
- — - —, then we have
- (1.33)
-
an entire function.
Therefore,
Let C = max {||hi1||k, ..., ||hiq||k} and
C(K) = max {C, ||hi||k (1.34)
Now combining equation (1.34) and 1.32) yields
– - (1.35)
(1.36) Now lett
-plane.
On repeating the above argument once more leads to –
-
(1.37) A-1 C2
-1 B C2 (k) || f ||2 (1.38)
Which
it suffices to show that
- -
- - (1.39)
- - -
(1.40)
- - - -
- - - (1.41)
Thus,
- - - -
- (1.42)
–
(1.43)
So, F is continuous everywhere. Proved.
and Finally statement [2] follows from equation (1.25) and equation (1.43), that is
- (1.44)
Statement [2] be also proved as follows.
Let the eigen vector of L form an orthonormal basis, hence they form an exact light frame with frame bounds = 1 and gn*s = gn. As a specified case, let = L2(I), where I = [a,b], -
(1.45)
If K is real, symmetric and in L2(Q), where Q = I x I, then L is a self-adjoint compact operator on L2(I). In addition, if the equation
(1.46)
basis for assumption of the theorem [1] if
(1.47) -
fixed point in I, then
- (1.48)
equation and in case the self adjoint boundary value problem it becomes the Green function of the problem.
1.4 Inversion formula
integral of F. For this purpose, we need to restrict the growth rate of the seque
sequence instead of natural numbers. For this let us assume that there exists a sequence of functions h that
- - (1.49)
0
And - (1.50)
1 if n = m Set
- - (1.51)
Hence - - (1.52)
Where X[- -
Let KN ( ) = - (1.53)
From equation (1.49) – (1.51) and parseval inequality, we have
- -
- (1.54)
But we know that
- (1.55)
Thus from equation (1.54) and (1.55) we obtain
- - -
- - (1.56)
-
-N < f, gn > gn* (1.57) Here the last inequality follows from the expression as
) < f, gn > gn*
Now interchanging the summation and integration sign is possibly by the dominated convergence theorem for Bochner integrals; since
- - - - -
-1 BD ||f||2 < (1.58)
- (1.59)
- equation (1.59) becomes
-
1.5 Sampling on unit sphere or 2-sphere S2
re 2-sphere, then (f) lm
—lm—(— —j— — (1.60)
with it. [First, we cannot sure about the numerical accuracy and the stability of this technique for a given number of samples. This is especially relevant if the results are going to fed to another sophisticated algorithms and the error accumulation may occur.
Second, if we attempt to improve accuracy by over sampling, then the computational and storage costs increase.
So, it is desirable to have more efficient and provably accurate techniques for signal processing and relevant computations on 2-sphere. Therefore, this has been an active and practically relevant area of research. Let widely used method has been proposed in that turns the above approximation into an exact computation for all degrees smaller than N/2, where N is the number of samples along the co-latitude and longitude. This is accomplished using a "quadrature" rule with approximately chosen weights. Here we briefly reviewed the results, as they can be useful in applications involving signal processing on 2-sphere].
1.5.1 Sampling distribution
Sampling on unit sphere or 2-sphere can be best understood using sampling distribution as well as sampling l > Lf. That is, there are a maximum of (Lf+1)2 non-zero spherical harmonic coefficients for f. Now any (Lf+1)2 independent samples as our sampling distribution (that is, (Lf+1)2 independent co-latitude and longitude pairs) lead to a linear system in the unknown spherical harmonic co-efficients, which can be solved exactly via Matrix inversion. The real issue have, though, is having a sampling distribution that leads to a numerically well-conditional system, match real world sampling schemes and more importantly leads to a reasonable computational complexity.
We can uniformly sample both the co- at point, that is
-1}
and (1.61)
-1}
obviously, there are more samples around the poles than the equator and this needs to be compensated for with some proper weighting. The specific weighted sampling distribution function have been proposed and corrected for minor scaling error, as
- - - - (1.62)
- -1} (1.63)
have a closed form solution as
-
-1} (1.64) 1.5.2 Sampling theorem on 2-sphere
The sampling theorem on 2- -sphere
2(Lf+1) uniform points on the sphere in both co-latitude and longitude according to the equation (1.61) and exactly revolved using.
N=- - —lm—(— —j— —k) (1.65)
-l, - possible aliasing will occur for degrees beyond Lf.
1.6 Error Calculation
Error calculations in terms of reliability and computational cost. Let us consider the samples of a signal f at
K ( ... N-
0.5481 0.4606 0.9703 0.4558 0.7807 0.9209 0.7523 0.5205
0.5589 0.8862 0.0155 0.0889 0.3017 0.8795 0.9923 0.7851 Where horizontal direction represents change in co-
Now using exact method or sampling theorem equation (6.65) on 2-sphere, that is
- - —lm—(— —j— —k)
where Y—lm—(— —j— —k) = (-1)m Yl- -
{-l, -l+1, ..., -l}. The vector of spherical coefficients have been found as f = (2.2829, 0.3304+i0.1439, 0.4692, -0.3304+i0.1439)', where ' represents transpose of vector matrix. Similarly, when we have used equation (6.60), that is
—lm—(— —j— — the vector of spherical coefficients have been found as f = (2.1809, 0.3945+i0.1499, 0.3909, -0.3945+i0.1499)'
when we compared the difference between exact method and approximate method, the difference is non- negligible. The normalized err, which is defined as
-l | (f)lm – -l | (f)lm |2 )1/2
Now when we use f as our reference to oversample the signal with N = 8 as -
Then we get the some spherical harmonic coefficient as in f, if we feed f2 in equation (6.65) with normalized error.
10-16.
We can also improve the accuracy of direct calculation f = (2.2535, 0.3305 + i0.1440, 0.4508, -0.3305 + method is superior to the direct computation method in terms of reliability and computational cost.
In evidence of the above statement let us compare the exact method (1.65) and the approximate method
= Y2- -
given below
-
-1}
r N = 8 as
In simulation, the error is defined as
-l | (f)lm – -1,m |2 )1/2
and from simulation the error has been calculated to be 2.1314 x 10-16 using exact method, whereas the error using approximate method with some number of sample N = 8 was found to be 1.7000 x 10-3.
Now when we increase the number of sample N'=32 which reduces the error to 5.8463 x 10-6.
2.1 Conclusion
Since sampling in a Hilbert space introduction in communication engineering by Shannon in 1948, the Whittaker-Shannon-Kolel'nikov sampling theorem has played an important role in both mathematics and electrical engineering. Hence an analog of the Wittaker-Shannon Kolel'nikov sampling theorem has been derived for functions with the values in a separable Hilbert space. One of the most important consequences of the derived sampling theorem is that it allows us to derive sampling theorems associated with boundary value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem.
As the aim of this thesis is to generalize the results obtained to derive a sampling theorem for vector valued functions. These functions take values in a separable Hilbert space . One of the interesting ramifications of this generalization is that it allows us to obtaining sampling theorems associated with boundary value problems and integral equations without restricting the sampling points to be the eigen values of the corresponding problem. In fact, the sampling points will be arbitrary, except for a restriction on their growth rate.
Some problems in sampling on 2-sphere has been considered to numerically compute the spherical harmonic coefficients of a given
entire 2-sphere and have been derived an approximate equation and compared it with sampling theorem on 2-sphere.
In article, we have compared the error results using approximate method and exact method. We came into conclusions that, the exact method is superior to direct method in terms of reliability and computation cost.
For example for a reference signal, whose speherical harmonic coefficient is known & considered sample points as 8 and calculated error which has been found 2.1314 x 10-16 in simulation on exact method, but it has been computed on approximate method the error has been found 1.7 x 10-3.
When the number of samples has increased from 8 to 32. Then the error has been reduced to 5.8463 x 10-6.
Thus number of sample also plays an important role to reduce the error. In second case, the normalized error has been found equal to 0.0662 when we have taken N = 4. When N has been increased 8 and compared error using exact and approximate method it has been concluded that normalized error has been reduced to 0.0145.
In summary, the exact method is superior to direct method of computation in terms of [a] Reliability performance and
[b] Computation cost 3.1 Future Scope
In order to find optimally concentrated signals in spatial or spectral domains has many applications in practice, which are similar in spirit and can be as diverse as applications of the concentration problem in time-frequency domain. For example, we may need to smooth a signal on the sphere to reduce the effects of noise or high spectral components by performing local convolutional averaging or filtering, which will be the topic of future scope for further research work as optimally concentrated window's that allow accurate joint spatio-spectral analysis of signals on the sphere.
4.1 References
[1.] Bachman. George; Narici. Lawrence; Beckenstein, Edward (2000), Fourier and wavelet analysis.
Universitext, Berlin, New York: Springer-Verlag, MR 1729490. ISBN 978-0-387-98899-3.
[2.] Bers. Lipman; John, Fritz; Schechter Martin (1981), Partial differential equation, American Mathematical Society, ISBN 0821800493[4] F. R. Kschischang, B. J. Prey, and H. A.
Loeliger, "Factor Graphs and the Sum-Product Algorithm", IEEE Transactions on Information Theory, vol.47. No.2,pp.498-519 February 2001.
[3.] I. Zayaed, Advances in Shannon's sampling theory, CRC Press, Boca Raton, FL(1993).
MR95f:9408.
[4.] Sampling theorems and bases in a Hilbert space. Information and Control, 4 (1961). pp. 97-117.
MR 26:4832.
[5.] Helmberg, G. (1969) Introduction to Spectral Theory in Hilbert Space. John Wiley & Sons, New York, NY (reprinted in 2008 by Dover Publications, Mineola, NY).
[6.] Huang, W., Khalid, Z., and Kennedy, R. A. (2011) "Efficient computation of spherical harmonic transform using parallel architecture ofCUDA," In Proc. Int. Con}. Signal Process. Commun. Syst., ICSPCS'2011, p. 6. Honolulu, HI.
[7.] Kennedy, R. A., Lamahewa, T. A., and Wei, L. (2011) "On azimuthally symmetric 2-sphere convolution," Digit. Signal Process., vol. 5, no. 11, pp. 660-666.
[8.] Kennedy, R. A., sadeghi, P., abhayapala, T. D., and jones, H. M. (2007) "Intrinsic limits of dimensionality and richness in random multipath fields," IEEE Trans. Signal Process., vol. 55, no.
6, pp. 2542-2556.
[9.] Reid, C. (1986) Hilbert- Courant. Springer-Verlag, New York, NY (previously published as two separate volumes: Hilbert by Constance Reid, Springer-Verlag, 1970; Courant in Gottingen and New York: The Story of an Improbable Mathematician by Constance Reid, Springer-Verlag, 1976).
[10.] Riesz, F. (1907) "Sur les systemes orthogonaux de functions," Comptes rendus de I'Academie des sciences, Paris, vol. 144, pp. 615-619.
[11.] Risbo, T. (1996) "Fourier transform summation of Legendre series and D-functions," J. Geodesy, vol. 70, no. 7, pp. 383-396.
[12.] Sadeghi, P., Kennedy, R. A., and Khalid, Z. (2012) "Commutative anisotropic convolution on the 2-sphere," IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6697-6703
